Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes

Abstract

In this paper, we study the Newton polytopes of $F$-polynomials in a TSSS cluster algebra $\mathcal{A}$ and generalize them to a larger set consisting of polytopes $N_h$ associated to vectors $h\in {\mathbb{Z}}^n$ as well as $\hat{\mathcal{P}}$ consisting of polytope functions ${\rho }_h$ corresponding to $N_h$. The main contribution contains that (i) obtaining a {\em recurrence construction} of the Laurent expression of a cluster variable in a cluster from its $g$-vector; (ii) proving the subset $\mathcal{P}$ of $\hat{\mathcal{P}}$ consisting of Laurent polynomials in $\hat{\mathcal{P}}$ is a strongly positive $\mathbb{Z}Trop(Y)$-basis for $\mathcal{U}(\A)$ consisting of certain universally indecomposable Laurent polynomials when $\A$ is a cluster algebra with principal coefficients. For a cluster algebra $\mathcal{A}$ over arbitrary semifield $\mathbb{P}$ in general, $\mathcal{P}$ is a strongly positive $\mathbb{Z}\P$-basis for the intermediate cluster subalgebra ${\mathcal{I}}_{\mathcal{P}}(\mathcal{A})$ of $\mathcal{U}(\mathcal{A})$. We call $\mathcal{P}$ the {\em polytope basis}; (iii) constructing some explicit maps among corresponding $F$-polynomials, $g$-vectors, $d$-vectors and cluster variables to characterize their relationship. Moreover, we give three applications of (i), (ii) and (iii) respectively.

68 pages

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